Vague groups

A vague set uses a truth grade of membership and a false grade of membership to indicate the positive evidence and the negative evidence, respectively. A vague set can overcome the problem that the grade of membership of a fuzzy set can not distinguish the two evidences from each other. However, the algebraic structures of vague sets are still not obtained. To solve the problem, this paper fuses vague sets and a class of fuzzy groups, presents the vague groups (VGs), puts forward and proves the following three theorems: 1) the vague homomorphism theorem; 2) vague Caushy lemma, and 3) vague Sylow theorem. In the structural properties of VGs, this paper solves the problems of the preservation of operations, the sameness of structures, and the relation between the number of order and the structure of VG. VGs also overcome the drawback of a class of fuzzy groups whose operations can not distinguish the positive evidence from the negative evidence.